Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-3x-4y &= 7 \\ -8x-8y &= 5\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $1$ $\begin{align*}6x+8y &= -14\\ -8x-8y &= 5\end{align*}$ Add the top and bottom equations. $-2x = -9$ Divide both sides by $-2$ and reduce as necessary. $x = \dfrac{9}{2}$ Substitute $\dfrac{9}{2}$ for $x$ in the top equation. $-3( \dfrac{9}{2})-4y = 7$ $-\dfrac{27}{2}-4y = 7$ $-4y = \dfrac{41}{2}$ $y = -\dfrac{41}{8}$ The solution is $\enspace x = \dfrac{9}{2}, \enspace y = -\dfrac{41}{8}$.